A look at Representations of SL(2,F_(q)) through the Lens of Size

• Main Authors: Gurevich, Shamgar, Howe, Roger
• Format: Journal Article
• Language: English
• Published: 23.05.2021
• Subjects: Mathematics - Representation Theory
• DOI: 10.48550/arxiv.2105.11008

How to study a nice function on the real line? The physically motivated Fourier theory technique of harmonic analysis is to expand the function in the basis of exponentials and study the meaningful terms in the expansion. Now, suppose the function lives on a finite non-commutative group G, and is invariant under conjugation. There is a well-known analog of Fourier analysis, using the irreducible characters of G. This can be applied to many functions that express interesting properties of G. To study these functions one wants to know how the different characters contribute to the sum? In this note we describe the G=SL(2,F_q) case of the theory we have been developing in recent years which attempts to give a fairly general answer to the above question for finite classical groups. The irreducible representations of SL(2,F_q) are "well known" for a very long time [Frobenius1896, Jordan1907, Schur1907] and are a prototype example in many introductory courses on the subject. We are happy that we can say something new about them. In particular, it turns out that the representations that were considered as "anomalous" in the "old" point of view (known as the "philosophy of cusp forms") are the building blocks of the current approach.